Cryptarithmetic

How to solve written letter as Squareroot of (PASSION) = KISS.
Each alphabet represents separate digit. I know the answer but i wish to know how to arrive at it by calculation without computer program.

First, write it as KISS x KISS = PASSION. We can also note that since PASSION is only 7 digits (<10,000,000), K can only be 1,2, or 3. Note also that as PASSION is square, the ones digit N can only be 0, 1, 4, 5, 6, or 9. Further, as N is the ones digit of SxS, N cannot be 0 of 5 (as that would require N=S). So N is 1, 4, 6, or 9. Each of these gives limited possibilities for S (N=1 -> S=9, N=4 -> S=2 or 8, N=6 -> S=4, N=9 -> S=3 or 7).
Try to proceed from there.

Okay Moo, you've made at least a couple of errors. Yes, while 6^2=36, so S cannot be 6, as then N=S; but 4^2=16, so N can be 6 if S is four.

Second, note that KISS is the root, and it is PASSION that is the square; thus if they are even, it is PASSION, and not (necessarily) KISS, that must be divisible by 4. (For example, the even square number 36 is divisible by four, but it's square root 6 is not).

Okay Moo, you've made at least a couple of errors. Yes, while 6^2=36, so S cannot be 6, as then N=S; but 4^2=16, so N can be 6 if S is four.

Second, note that KISS is the root, and it is PASSION that is the square; thus if they are even, it is PASSION, and not (necessarily) KISS, that must be divisible by 4. (For example, the even square number 36 is divisible by four, but it's square root 6 is not).

First, write it as KISS x KISS = PASSION. We can also note that since PASSION is only 7 digits (<10,000,000), K can only be 1,2, or 3. Note also that as PASSION is square, the ones digit N can only be 0, 1, 4, 5, 6, or 9. Further, as N is the ones digit of SxS, N cannot be 0 of 5 (as that would require N=S). So N is 1, 4, 6, or 9. Each of these gives limited possibilities for S (N=1 -> S=9, N=4 -> S=2 or 8, N=6 -> S=4, N=9 -> S=3 or 7).
Try to proceed from there.

--Kevin C.

Thank you.

SS can be either 22, 33, 44, 77 or 99. I will write about how to arrive at this soon.